www.iap.uni-jena.de

Medical Photonics Lecture

Optical Engineering

Lecture 13: Metrology

2019-07-10

Herbert Gross

Summer term 2019

2

Contents

No Subject Ref Date Detailed Content

1 Introduction Zhong 10.04.Materials, dispersion, ray picture, geometrical approach, paraxial approximation

2 Geometrical optics Zhong 17.04.Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant

3 Diffraction Zhong 24.04.Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function

4 Components Kempe 08.05. Lenses, micro-optics, mirrors, prisms, gratings

5 Optical systems Zhong 15.05.Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting

6 Aberrations Zhong 22.05. Introduction, primary aberrations, miscellaneous

7 Image quality Zhong 29.05. Spot, ray aberration curves, PSF and MTF, criteria

8 Instruments I Kempe 05.06.Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes

9 Instruments II Kempe 12.06.Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts

10 Instruments III Kempe 19.06.Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes

11 Photometry Zhong 26.06.Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory

12 Illumination systems Gross 03.07.Light sources, basic systems, quality criteria, nonsequential raytrace

13 Metrology Gross 10.07. Measurement of basic parameters, quality measurements

Collimated incident light

Calibrated collimator with focal length fc and test chart with size y

Selection of sharp image plane

Analysis of image size

Measurement of Focal Length with Collimator

f'c f'

test chart image

y y'

collimator test optic

y

yff c

'''

Setup with distance object-image L > 4f

Known location of the principal plane P of the system

distance dP between principal planes

Selection of two system locations with sharp image

Relative axial shift D between the two setups

Measurement of Focal Length According to Gauss

H

H

dL

DdLf

44

2

F F'

L

F F'

dP

Ds

s'

P P'

P P'

position 1

position 2

f f'

Telecentric movable measurement microscope with offset y

Focusing of two different test charts with sizes y1 and y2

Determination of the focal length by

Measurement of Focal Length with Abbe-Focometer

test lens

measuring

microscope

y1

Fy2

e y

f focusing

movement

u

test

scale 1

test

scale 2

e

yy

f

yu 12tan

Setup with fiber and plane mirror for autocollimation

Change of distance between test lens and fiber

Analysis of the recoupled power into the fiber (confocal) gives the focal point

Measurement of Focal Length by Confocal Setup

z

lens under

testplane mirror

autocollimation

case

recoupled

energy

lens position z

Afocal setup with sharp image plane

Measurement of long focal lengths

Insertion of test system in collimated light segment and refocussing

Applying the lens makers formula

Measurement of Focal Length with Focimeter

reference

image

plane

test

target

lens

under

test

f2f2

new image

plane

x

collimator

lens

focusing

lens f2

re-focusing

2

22

11

f

x

ff

Setup with collimator and two Ronchi rulings

System under test is inserted

Grating period d and azimuthal angle q between the gratings

Moire pattern is rotated by angle a, if test lens acts as focussing element

Radius of curvature R or focal length

Measurement of Focal Length by Moire Deflectometry

test

target

lens

under

test

observation

plane

collimator

lens

Ronchi

gratings

a

qMoire

pattern

d

aq tan

dR

Setup with Ronchi grating in collimated light gives a series of Talbot images

The Talbot planes are imaged by the system under test

Analysis of the imge plane by lens formula gives the desired focal length

By use of several planes, the position of the principal plane can be eliminated

A second Ronchi grating can be used to find the accurate image planes

Measurement of Focal Length by Talbot Imaging

test

target

lens

under test

intermediate

Talbot images

collimator

lens

Ronchi

grating

-2 -1 0 +1 +2 +2 +1 0 -1 -2

Talbot

images

Criteria for best focus:

1. Paraxial centre of curvature for the paraxial spherical wave of an on axis object point.

2. Maximum of the Strehl ratio

3. Smallest rms-value of the wave aberration

4. Highest contrast of the modulation of an object feature of given spatial frequency

5. Highest value of the slope of an edge

6. Highest value of the entropy of the detected digital image

Requirements for focus detection procedure

1. Steep curve dependency to get high accuracy

2. Robust definition to deliver a large dynamic range

3. Suppression of side lobe effects to guarantee an unambiguous solution

4. High frequency pre-filtering to be noise insensitive

Determination of Best Focus

Blur of defocussed

plane

Minimum of image entropy

Maximum of image contrast

Determination of Best Focus

object

plane

optical

systembest

focus

plane

actual

plane

defocus

Dz

D blur

diameter

j

jj wwE 2log

Phase analysis by Zernike

coefficient c4

Measurement with two

Ronchi gratings

Determination of Best Focus

2

44

1NAz

nc D

object

plane

optical

systemmeasuring

planesensor

plane

defocus

Dz

Ronchi

gratings

cj

z grating

location

0

0

c4

c9

Measurement by image abalysis:

1. Maximum gradient of edges

2. Power of gradients

3. Laplacian

Determination of Best Focus

-4 -3 -2 -1 0 1 2 3 40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

rms ( I )

Dz

[a.u.]

22

),(

y

I

x

IyxIg

dydxyxIG2

),(

dydxyxIL2

2 ),(

Measurement for systems in air via the nodal planes

Imaging of a test pattern with a collimator onto a detector

Invariant lateral image location for rotated system around the nodal point

Critical: vignetting effects for large angles

Measurement of Principal Planes

test

patterncollimator

test system detector

P P'

tilt

angle

j

invariant

image

position

turning

point

movable along z

Setup of the test lens with different object locations: axial shift D

Analysis of the lens imaging formula

Minimazing the error of several measurements j

Measurement of Principal Planes

test

patterntest system detector

P P'

sj s'j

aj

a'jD

reference

plane

faa jj

1

'

11

D

D

2''' DD jjjjjj aaaaaaf

Setup with collimating auxiliary lens

Determination with measuring microscope (dynameter)

Measurement of Pupil Size

pinhole

illuminationlens under

test

image

plane

auxiliary

lens with fglass plate

with scale

u

f

D

movable for

collimation

collimated

f

Du

2tan

light

sourcecollimator test system

movable

dynameter lensexit

pupil

Setup with Ronchi grating

Measurement of the lateral shift of higher diffraction orders at distance z

High-NA in microscopy: NA>1

Measurement of total internal reflection

of fluorescence light

Measurement of Pupil Size

test

system

image

plane

f

Ronchi

grating

D

Dx

0th order

+1st order

-1st order

z

DExP

a

x

Dxq

Fluorescence intensity

1.5 1.0 0.5 0 0.5 1.0 1.5

rmax

rTIR

pupil illumination

TIR

NA

Measurement of object sided telecentricity errors by lateral shift of image location during

defocussing

High accuracy measurement by interferometry and measurement of Zernike coefficients

c2/3.

Measurement of Telecentricity

object

space

2DzF

image

plane

exact

telecentric

telecentricity

error

jT

Dx'

Measurement of reflexes at lens vertex points

Analysis of confocal signal in autocollimation

Avoiding spherical aberration induced errors by ring illumination

Measurement of Lens Position

confocal

pinhole

+M

lens under test

sensor

movable for

focusing on one

surface vertex

lenschannel 1

channel 2

laser

source

beam splitter

ring

stop

vertex

Sj

z

DI

confocal

difference signal

S1 S3 S4S2

d1

confocal pinhole -M

Measurement of tilt errors (plane or spherical surface) in autocollimation

Projection of the cross

Observation of lateral shift in Fourier plane

Autocollimation

2x f j

eyepiece

illumination

objectiv

collimator

focal length freticle

measurement

scale

test

object

centered

tilt angle j

beam

splitter

f'

x

j

Measurement og 90° angle in air

Wedge interferogram

Kink of fringes gives angle error

Test of Prism Angles

D

m

2

a

incident

collimated

beam D

test piece

auxiliary

glass plate

in contact

test range DP

measuring

angle

90a

test angle

90a

emerging

wavefront

a

2a

a

d

/ 2

s

deviation of the

fringes

distance of

fringes

basis D / 2

Projection of test marker

Autocollimation of sharp image, focal point coincides with center of curvature of surface

with radius r

Rotation of test system: tilt of surface induces a

lateral shift of the image

Problems with inner surfaces

Measurement of Centering Errors by Reflected Light

rvv M 2

q

illumination

zoom systemlight

source

adjustment of

the image

location

detector

deviation circle

test surface

C

r

tilt

rotating

Thin collimated beam through lens

Focussing of the beam onto detector

Measurement of wedge angle by lateral shift v

Tilt angle of lens not detectable

Not feasible for very short focal lengths

Measurement of Centering Errors in Transmission

f

vnn 21)1()1( aaqj

q

laser

light

source

lens under test

focal length f

wedge error

detector

circle of

deviation

objective

j

inclination

of the axis

rotated

Reasons for reduced system transmission:1. Absorption in the bulk material of the components

2. Scattering in the bulk materials by inclusions or finite scattering parameters

3. Absorption in the coatings of the surfaces

4. Partial reflection or transmission at the coatings at transmissive or reflective surfaces

5. Blocking of light via mechanical or diaphragm parts of the system due to vignetting

6. Scattering of light by local surface imperfections or non-perfect polished surfaces

7. Deflection of light by diffraction of the light at edges

8. Deflection of light in unwanted higher orders of diffractive elements

Usually strong dependency on:

1. field position

2. wavelength of light

3. used pupil location

4. polarization

Critical:

1. absolute values for test lens

2. influence of auxiliary components

3. change of vignetting and incidence angles

Measurement of Transmission

Measurement of transmission:

Reasons for measurement errors:1. Absorption in the component materials

2. Absorption in the coatings

3. Finite reflectivity of the coatings

4. Vignetting of the aperture bundle for oblique chief rays

5. Natural vignetting according for oblique chief rays and projection of tilted planes

6. False light from surrounding light sources, which reach the image plane

7. Scattering of light at components of the system mechanical design

8. False light due to ghost images or narcissus in infrared systems

Measurement of Transmission

a) Calibration setup b) Measurement setup

source

collimator

Ulbricht

sphere

mirror

source

collimator

system under test

Ulbricht

spherePin Pin

Pout

out

in

P

PT

Measurement of unwanted light: 2 different approaches:

1. object area black, surrounded by bright source

detection of irradiance in image region

2. intensive isolated point light source in the object plane at different locations

detection of artificial distributions in the image area: glare spread function

Measurement of Ghost Images and Veiling Glare

system under testbright

illumination

field

black

screen

image

plane

irradiance

not

perfectly

black

Tactile Measurement

Ref: H. Hage / R.Börret

Scanning method

- Sapphire sphere probes shape

- slow

- only some traces are measured

Universal coordinate measuring machine

(CMM) as basic engine

Contact can damage the surface

Accuracy 0.2 mm in best case

27

Influence of the tast sphere:

filtering of higher spatial frequencies due to finite size

Gradient of skew touching geometry can be calibrated and corrected

Periods of same size as the sphere: error in amplitude

High frequencies: not detected

Tactile Measurement

A

high accuracy

v

gradient

correction

amplitude

reduced

period not

resolved

calibration

according to

contact slope

29

Measurement by Fringe Projection

Projection of a light sheet onto

a deformed surface

Corresponds to one fringe in

more complicated pattern

projection

30

Example Fringe projection

Monochromatic illuminated technical surface

31

3D Shape Measurement for Biometry

Colored biometric fringe projection

Projection of a 2D triangular pattern

Shape measurement of a surface

Projection of a fringe pattern onto the surface

Observation of the fringe deformation by a camera

A shift corresponds to a change in depth

Non-trivial image processing

Shape Measurement by Fringe Projection

light

source

collector

amplitude

grating

condenser

surface

under test

sensor

objective for

detection

Data evaluation

Fringe period

Lateral shift

Corresponding depth value

z

1cosq

dd x

21 coscos qq zu

2

21

1

21

cos

sin

cos

coscos)(

q

qq

q

qq

d

z

d

z

d

uxf

x

Fringe Projection

test surface

P

P

1

2

z

u

x

z

q2q

1

projected fringes

d

dx

Projection of a narrow beam onto a surface

Backscattering of light

Observation un der an oblique angle:

Lateral shift of image point corresponds to depth

Dz

Scattered light amplitude depends strongly on material

properties

Problems with polished surfaces

u

Cz

sinsin2

D

q

Triangulation

q

test piece

u

sensor

illumination

observation

Dx

Dz

laser

Measurement of the refractive index of a liquid

Thin film of test liquid between prisms,

adjustment of total internal reflection

Special setup with direct sight prisms, no color fringes

Abbe Refractometer

eyepieceobjectivecompensators

Amici prism without

deviation

measuring

prism

test liquid

telescope

image

direction of the

telescope axis

j

n

a

Measurement of material data by polarization

Incident linear polarized light

Reflected light elliptical polarized

Compensation of ellipticity, quantitative determination of null-test parameter

Ellipsometry for Polarization Measurement

x

yz

a1

light

source

j

test

piece

linear

polarization

elliptical

polarization

a2

receiver

linear

polarization

compensator

analyzer

variabel

variabel

Two beam interference of two waves:

- propagation in the same direction

- same polarization

- phase difference smaller than axial length of coherence

Coherent superposition of waves

Difference of phase / path difference

Number of fringes

location of same phase

Conrast

122121

2

21

cos2 jD

IIII

EEI

122

j

DDs

j sN

D

D

2

12

21

21

minmax

minmax2

II

II

II

IIK

Two Beam Interference

Testing with Fizeau Interferometer

Long common path, quite insensitive setup

Autocollimating Fizeau surface quite near to test surface, short cavity length

Imaging of test surface on detector

Straylight stop to bloc unwanted light

Curved test surface: auxiliary objective lens (aplanatic, double path)

Highest accuracy

detector

beam

splitter

collimatorconvex

surface

under test

light

source

Fizeau

surface

auxiliary lens

stop

Testing with Twyman-Green Interferometer

Short common path,

sensible setup

Two different operation

modes for reflection or

transmission

Always factor of 2 between

detected wave and

component under test

detector

objective

lens

beam

splitter 1. mode:

lens tested in transmission

auxiliary mirror for auto-

collimation

2. mode:

surface tested in reflection

auxiliary lens to generate

convergent beam

reference mirror

collimated

laser beam

stop

test surface

beamsplitter

reference surface

here: flat

illumination

to detector

path difference

mRrm

Test by Newton Fringes

Reference surface and test surface with nearly the same radii

Interference in the air gap

Reference flat or curved possible

Corresponds to Fizeau setup

with contact

Broad application in simple

optical shop test

Radii of fringes

40

Ref: W. Osten

Interferograms of Primary Aberrations

Spherical aberration 1

-1 -0.5 0 +0.5 +1

Defocussing in

Astigmatism 1

Coma 1

41

Real Measured Interferogram

Problems in real world measurement:

Edge effects

Definition of boundary

Perturbation by coherent

stray light

Local surface error are not

well described by Zernike

expansion

Convolution with motion blur

Ref: B. Dörband

42

Interferogram - Definition of Boundary

Critical definition of the interferogram boundary and the Zernike normalization

radius in reality

43

Interferometry

Color fringes of a broadband interfergram

Ref: B. Dörband

Separation of wavefront:

self reference

Interferograms are looking completly different

Aperture reduced due to shear

Splitting and shift of wavefront:

- by thin plate

- by grating

d

Shearing Interferometer

source

shear

distance

Focussing onto a transparent plagte with pinhole

Pinhole creates a reference spherical wave

Optimization of contrast:

- size of pinhole

- numericalaperture

- transparency of the plate

Very stable setup

transparent plate

with pinhole

wavefront

under test reference

wavefront

Point Diffraction Interferometer

tDD /1

t

tA

D

D

sin)(

deAtE ti2)()(

Temporal Coherence

I()

Radiation of a single atom:

Finite time Dt, wave train of finite length,

No periodic function, representation as Fourier integral

with spectral amplitude A()

Example rectangular spectral distribution

Finite time of duration: spectral broadening D,

schematic drawing of spectral width

Corresponding axial coherence time

Axial coherence length

47

D

1c

cc cl

48

Lateral and Axial Resolution

Intensity distributions

Aberration-free Airy pattern:

lateral resolution

axial resolution

lateral axial

Ref: U. Kubitschek

NADAiry

22.1

2NA

nRE

OCT Setup

Basic principle of OCT

Michelson interferometer

Time domain signal

receiverfirst mirrorfrom

source

signal

beam

reference

beam

beam

splitter

second

mirror

moving

overlap

lc

z z

relative

moving

I(z)

wave trains

with finite

length

-4 -2 0 2 4

0

0,2

0,4

0,6

0,8

1,0

-4 -2 0 2 4

0

0,2

0,4

0,6

0,8

1,0

primary

signal

filtered

signal

t

49

Example:

sample with two reflecting surfaces

1. Spatial domain

2. Complete signal

3. Filtered signal

high-frequency content removed

50

Optical Coherence Tomography

Ref: M. Kaschke

Example of OCT Imaging

Example:

Fundus of the human eye

51

Dimensions of OCT imaging:

a) only depth (A-scan), one-dimensional

b) depth and one lateral coordinate B-scan), two-dimensional

c) all three coordinates, volume imaging

52

Optical Coherence Tomography

Ref: M. Kaschke

53

White Light Interferometry

Examples

Ref: R. Kowarschik

Moving a knife edge perpendicular through

the beam cross section

Relationship between power

transmission and intensity:

Abel transform for circular

symmetry

Example: geometrical spot with spherical aberration

dr

drrrIxP

x

22

)(2)(

z

beforecaustic

zone rays belownear paraxial

focus

Knife Edge Method

y

x

beam

x

knife edge

movement

54

Measurement of an edge image

Evaluating the derivative:

Line spread function

Fourier transform:

optical transfer function

MTF-Measurement by Edge Spread Function

'

)'()'(

xd

xIdxI ESF

LSF

)'(ˆ)( xIFsH LSFOTF

x

edge

object

incoherent

illumination

Iin(x)

detector

plane

x'diffracted

light

geometrical

image

of the edge

I(x')

d I(x')

dx'

55

Setup:

Imaging of a grating

Possible realizations:1. Density type grating, the sine wave is modelled by gray levels

2. Area type gratings, the sine wave is modelled by geometrical sine-shaped structures

Area coded sine grating:

MTF-Measurement by Imaging Gratings

lamp

sensor

slitlens under

test

diffusor

grating

object

Hartmann Shack Wavefront Sensor

detector

xDmeas

u

hf

array

spot

offset

Darray

Dsub

refractive

index n

wave-

front

Lenslet array divides the wavefront into subapertures

Every lenslet generates a simgle spot in the focal plane

The averaged local tilt produces a transverse offset of the spot center

Integration of the derivative matrix delivers the wave front W(x,y)

57

Typical setup for component testing

Lenslet array

Hartmann Shack Wavefront Sensor

subaperture

point spread

function

2-dimensional

lenslet array

a) b)

test surface

telescope for adjust-

ment of the diameter

detector

collimator

lenslet

array

fiber

illumination

beam-

splitter

Spot Pattern of a HS - WFS

a) spherical aberration b) coma c) trefoil aberration

Aberrations produce a distorted spot pattern

Calibration of the setup for intrinsic residual errors

Problem: correspondence of the spots to the subapertures

Real Measurement of a HS-WFS

Problem in practice: definition of the boundary

Separated spots in case of diffraction

s

obj

s

gesamt

sd

ddf

Dddd

d

dd

44.2' 2121

1

2)(

Hartmann Method

Hartmann

diaphragm

ideal

image

plane

2. camera

planeobject

plane

geometrical

projected size

of pinhole

finite site

source field

of view

broadening

due to

diffraction

spot

intensity

pinhole

defocus and

diffraction

convolution

with field of

view

total

d1

f

Dobj

d's

DAiry

/2

D'Feld

/2

ds

h

d2

Hartmann Method

Real pinhole pattern with signal

Problems with cross talk and threshold

L

threshold

peak

minimum

c)

D

d a

b

a) b)